CMB, The Horizon Problem and a comment on BH's

[superg33k]

Just to keep everything on track, my question is:

Given that the universe initially had "a small initial radius." were particles that now casually disconnected regions of the CMB within each others particle horizon? And if not, is it because they were separating at a speed close to, or greater than, c? (separating by expansion)

 

[bapowell]

How do you conclude this from the first statement?

Nice catch. I forgot the important assumption: in a spacetime in which physical length scales, $l_{\rm phys} \propto a(t)$ (where $a(t)$ is the scale factor) expand (monotonically) at a slower rate than the Hubble radius, $d_{H} \propto H^{-1}$, i.e.
$$\frac{d}{dt}\left(\frac{d_{H}}{l_{\rm phys}}\right) > 0$$
and if the physical separation between two points a and b at a time t* satisfies
$$l_{\rm phys}(t^*) > d_H(t^*)$$
then $l_{\rm phys}(t) > d_H(t)$ for all $t < t^*$. From
$$\frac{d}{dt}\left(\frac{d_{H}}{l_{\rm phys}}\right) = \frac{d}{dt} \left(\frac{1}{aH}\right) = -\frac{\ddot{a}}{\dot{a}^2} > 0$$
we see that our assumption holds in all spacetimes for which $\ddot{a}<0$ -- non-inflationary spacetimes.

 

[superg33k]

Ok, the texts I have read didn't go into the horizon problem that mathsy but I think I got it. From what I gather you are saying that all points that are outside each others Hubble radius now have always been outside each others Hubble radius.

To apply this to the CMB I am going to assume that the 2 separate parts are outside each others Hubble radius which is why they are in casually disconnected regions of space. Thus they have always been outside each others Hubble radius. So they have been separating faster than c since the big bang?

 

[bapowell]

Ok, the texts I have read didn't go into the horizon problem that mathsy but I think I got it. From what I gather you are saying that all points that are outside each others Hubble radius now have always been outside each others Hubble radius.

Yes, for non-inflationary spacetimes, this is correct.

To apply this to the CMB I am going to assume that the 2 separate parts are outside each others Hubble radius which is why they are in casually disconnected regions of space. Thus they have always been outside each others Hubble radius. So they have been separating faster than c since the big bang?

Yes.

 

[superg33k]

Excellent. Thanks for going through this all with me. I'm happy to say the horizon problem makes sense. Additionally I never realized that non-infamitory models required a constant rate of expansion of the scale factor, which makes sense also.

 

[bapowell]

Excellent. Thanks for going through this all with me. I'm happy to say the horizon problem makes sense.

Great! Happy to help.

Additionally I never realized that non-infamitory models required a constant rate of expansion of the scale factor, which makes sense also.

I hope I didn't say this! By definition, non-inflationary spacetimes need only satisfy $\ddot{a}<0$. The scale factor in non-inflationary models is not necessarily constant -- for example, in a radiation dominated universe $a(t) \sim t^{1/2}$, whereas in a matter dominated one $a(t) \sim t^{2/3}$.

 

[superg33k]

I hope I didn't say this!

Yep, your right, you didn't. An inequality became an equality in my head.